Question

Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE ?

• A. If $n=8$, then $(\sqrt{2}–1)r<R$
• B. If $n=4$, then $(\sqrt{2}–1)r<R$
• C. If $n=12$, then $\sqrt{2}(\sqrt{3}+1)r>R$
• D. If $n=5$, then $r<R$

Answer 1

Answer: (A), (C)

If $n=12$, then $\sqrt{2}(\sqrt{3}+1)r>R$



$\sin \left(\frac{\pi }{n}\right)=\frac{r}{R+r}$
$⟹\frac{R}{r}+1=\text{cosec}\left(\frac{\pi }{n}\right)$

$⟹R=r[\text{cosec}\left(\frac{\pi }{n}\right)-1]$

$\left(A\right)n=4,R=r(\sqrt{2}–1)$

$\left(B\right)n=5,R=r[\text{cosec}\left(\frac{\pi }{5}\right)-1]$

$⟹R<r[\text{cosec}\left(\frac{\pi }{6}\right)-1]⟹R<r$

$\left(C\right)n=8,R=r[\text{cosec}\left(\frac{\pi }{8}\right)-1]$

$⟹R>r[\text{cosec}\left(\frac{\pi }{4}\right)-1]$

$\left(D\right)n=12,R=r[\text{cosec}\left(\frac{\pi }{12}\right)-1]$

$⟹R=[\sqrt{2}(\sqrt{3}+1)-1]r$

$⟹R<\sqrt{2}(\sqrt{3}+1)r$